Upload
gabriel-sydney-wilkins
View
221
Download
0
Embed Size (px)
Citation preview
Lesson 4 -Part AForecasting
• Quantitative Approaches to Forecasting
• Components of a Time Series
• Measures of Forecast Accuracy
• Smoothing Methods• Trend Projection
• Trend and Seasonal Components
• Regression Analysis
Forecasting Methods
ForecastingForecastingMethodsMethods
ForecastingForecastingMethodsMethods
QuantitativeQuantitativeQuantitativeQuantitative QualitativeQualitativeQualitativeQualitative
CausalCausalCausalCausal Time SeriesTime SeriesTime SeriesTime Series
SmoothingSmoothingSmoothingSmoothing TrendTrendProjectionProjection
TrendTrendProjectionProjection
Trend ProjectionTrend ProjectionAdjusted forAdjusted for
Seasonal InfluenceSeasonal Influence
Trend ProjectionTrend ProjectionAdjusted forAdjusted for
Seasonal InfluenceSeasonal Influence
Quantitative Approaches to Forecasting• Quantitative methods are based on an analysis of historical data
concerning one or more time series.
• A time series is a set of observations measured at successive points in time or over successive periods of time.
• If the historical data used are restricted to past values of the series that we are trying to forecast, the procedure is called a time series method.
• If the historical data used involve other time series that are believed to be related to the time series that we are trying to forecast, the procedure is called a causal method.
Time Series Methods• Three time series methods are:
– smoothing– trend projection– trend projection adjusted for seasonal influence
Components of a Time Series• The pattern or behavior of the data in a time series has
several components.
• The four components we will study are:
TrendTrendTrendTrend CyclicalCyclicalCyclicalCyclical SeasonalSeasonalSeasonalSeasonal IrregularIrregularIrregularIrregular
Components of a Time Series
• Trend Component
– The trend component accounts for the gradual shifting of the time series to relatively higher or lower values over a long period of time.
- Trend is usually the result of long-term factors such as changes in the population, demographics, technology, or consumer preferences.
Components of a Time Series
– Any regular pattern of sequences of values above and below the trend line lasting more than one year can be attributed to the cyclical component.
• Cyclical Component
– Usually, this component is due to multiyear cyclical movements in the economy.
Components of a Time Series
• The The seasonal componentseasonal component accounts for accounts for regular patterns of variability within certain regular patterns of variability within certain time periods, such as a year.time periods, such as a year.
• Seasonal Component
• The variability does not always correspond The variability does not always correspond with the seasons of the year (i.e. winter, with the seasons of the year (i.e. winter, spring, summer, fall).spring, summer, fall).• There can be, for example, within-week or There can be, for example, within-week or within-day “seasonal” behavior.within-day “seasonal” behavior.
Components of a Time Series
• The The irregular componentirregular component is caused by short- is caused by short-term, unanticipated and non-recurring term, unanticipated and non-recurring factors that affect the values of the time factors that affect the values of the time series.series.
• Irregular Component
• It is unpredictable.It is unpredictable.
• This component is the residual, or “catch-This component is the residual, or “catch-all,” factor that accounts for unexpected all,” factor that accounts for unexpected data values.data values.
Measures of Forecast Accuracy
The average of the squared forecast errors for the historical data is calculated. The forecasting method or parameter(s) that minimize this mean squared error is then selected.
The mean of the absolute values of all forecast errors is calculated, and the forecasting method or parameter(s) that minimize this measure is selected. (The mean absolute deviation measure is less sensitive to large forecast errors than the mean squared error measure.)
• Mean Squared Error
• Mean Absolute Deviation
Smoothing Methods
• Three common smoothing methods are:
• In cases in which the time series is fairly stable and has no significant trend, seasonal, or cyclical effects, one can use smoothing methods to average out the irregular component of the time series.
Exponential SmoothingExponential Smoothing
Weighted Moving AveragesWeighted Moving Averages
Moving AveragesMoving Averages
Smoothing Methods
The moving averages method consists of computing an average of the most recent n data values for the series and using this average for forecasting the value of the time series for the next period.
(most recent data values)Moving Average =
n
n(most recent data values)
Moving Average = n
n
• Moving Averages
Sales of Comfort brand headache medicine forSales of Comfort brand headache medicine for
the past ten weeks at Rosco Drugsthe past ten weeks at Rosco Drugs
are shown on the next slide. If are shown on the next slide. If
Rosco Drugs uses a 3-periodRosco Drugs uses a 3-period
moving average to forecast sales,moving average to forecast sales,
what is the forecast for Week 11?what is the forecast for Week 11?
• Example: Rosco Drugs
Smoothing Methods: Moving Averages
• Example: Rosco Drugs
Smoothing Methods: Moving Averages
1122334455
66778899
1010
110110115115125125120120125125
120120130130115115110110130130
WeekWeek WeekWeekSalesSales SalesSales
Time Series Plot using MINITAB
Week
Sale
s
10987654321
140
130
120
110
100
90
80
70
60
50
Time Series Plot of Sales
•Time series appears to be stable over time, though random variability is present.•Thus, smoothing methods are applicable.
Smoothing Methods: Moving Averages
11223344556677889910101111
110110115115125125120120125125120120130130115115110110130130
116.7116.7120.0120.0123.3123.3121.7121.7125.0125.0121.7121.7118.3118.3118.3118.3
123.3123.3121.7121.7125.0125.0121.7121.7118.3118.3118.3118.3
116.7116.7120.0120.0
WeekWeek SalesSales 3MA3MA ForecastForecast
(110 + 115 + (110 + 115 + 125)/3125)/3
(110 + 115 + (110 + 115 + 125)/3125)/3
Moving Averages using MINITAB
You can specify this i.e. 4, 6, 12, … & its depend on the problem
# of future forecasts needed
# of data points
Moving Average for Sales
Moving Average: Length 3
Time Sales MA Predict Error
1 110 * * *
2 115 * * *
3 125 116.667 * *
4 120 120.000 116.667 3.3333
5 125 123.333 120.000 5.0000
6 120 121.667 123.333 -3.3333
7 130 125.000 121.667 8.3333
8 115 121.667 125.000 -10.0000
9 110 118.333 121.667 -11.6667
10 130 118.333 118.333 11.6667
Forecasts
Period Forecast 95% CI
11 118.333 101.954 134.713
Week
Sale
s
1110987654321
135
130
125
120
115
110
105
100
Moving Average
Length 3
Accuracy MeasuresMAPE 6.3203MAD 7.6190MSD 69.8413
Variable
Forecasts95.0% PI
ActualFits
Moving Average Plot for Sales
MSE
# of values to be included in the moving average is chosen such that itminimizes the MSE
Smoothing Methods• Weighted Moving Averages
– The more recent observations are typically given more weight than older observations.
– For convenience, the weights usually sum to 1.
– To use this method we must first select the number of data values to be included in the average.
– Next, we must choose the weight for each of the data values.
Smoothing Methods
• Weighted Moving Averages
– An example of a 3-period weighted moving average (3WMA) is:
3WMA = .2(110) + .3(115) + .5(125) = 1193WMA = .2(110) + .3(115) + .5(125) = 119
Most recent of Most recent of thethe
three three observationsobservations
Weights (.2, .3,Weights (.2, .3,and .5) sum to and .5) sum to
11
Smoothing Methods
• Exponential Smoothing– This method is a special case of a weighted moving
averages method; we select only the weight for the most recent observation.
– The weights for the other data values are computed automatically and become smaller as the observations grow older.
– The exponential smoothing forecast is a weighted average of all the observations in the time series.
Smoothing Methods• Exponential Smoothing Model
FFtt+1+1 = = YYtt + (1 – + (1 – ))FFtt
wherewhere
FFtt+1+1 = forecast of the time series for period = forecast of the time series for period tt + 1 + 1
YYtt = actual value of the time series in period = actual value of the time series in period tt
FFtt = forecast of the time series for period = forecast of the time series for period tt
= smoothing constant (0 = smoothing constant (0 << << 1) 1)
To start theTo start thecalculations,calculations,we let we let FF11 = =
YY11
To start theTo start thecalculations,calculations,we let we let FF11 = =
YY11
– With some algebraic manipulation, we can rewrite Ft+1 = Yt + (1 – )Ft as:
Smoothing Methods
• Exponential Smoothing Model
FFtt+1+1 = = FFtt + + YYtt – – FFtt))
– We see that the new forecast Ft+1 is equal to the previous forecast Ft plus an adjustment, which is times the most recent forecast error, Yt – Ft.
Sales of Comfort brand headache medicine forSales of Comfort brand headache medicine for
the past ten weeks at Rosco Drugsthe past ten weeks at Rosco Drugs
are shown on the next slide. Ifare shown on the next slide. If
Rosco Drugs uses exponentialRosco Drugs uses exponential
smoothing to forecast sales, whichsmoothing to forecast sales, which
value for the smoothing constant value for the smoothing constant ,,
.1 or .8, gives better forecasts?.1 or .8, gives better forecasts?
• Example: Rosco Drugs
Smoothing Methods: Exponential Smoothing
• Example: Rosco Drugs
1122334455
66778899
1010
110110115115125125120120125125
120120130130115115110110130130
WeekWeek WeekWeekSalesSales SalesSales
Smoothing Methods: Exponential Smoothing
Smoothing Methods: Exponential Smoothing
• Exponential Smoothing ( = .1, 1 - = .9)
F1 = 110
F2 = .1Y1 + .9F1 = .1(110) + .9(110) = 110
F3 = .1Y2 + .9F2 = .1(115) + .9(110) = 110.5
F4 = .1Y3 + .9F3 = .1(125) + .9(110.5) = 111.95F5 = .1Y4 + .9F4 = .1(120) + .9(111.95) = 112.76
F6 = .1Y5 + .9F5 = .1(125) + .9(112.76) = 113.98
F7 = .1Y6 + .9F6 = .1(120) + .9(113.98) = 114.58
F8 = .1Y7 + .9F7 = .1(130) + .9(114.58) = 116.12
F9 = .1Y8 + .9F8 = .1(115) + .9(116.12) = 116.01
F10= .1Y9 + .9F9 = .1(110) + .9(116.01) = 115.41
Smoothing Methods: Exponential Smoothing• Exponential Smoothing ( = .8, 1 - = .2)
F1 = 110
F2 = .8(110) + .2(110) = 110
F3 = .8(115) + .2(110) = 114
F4 = .8(125) + .2(114) = 122.80F5 = .8(120) + .2(122.80) = 120.56
F6 = .8(125) + .2(120.56) = 124.11
F7 = .8(120) + .2(124.11) = 120.82
F8 = .8(130) + .2(120.82) = 128.16
F9 = .8(115) + .2(128.16) = 117.63
F10= .8(110) + .2(117.63) = 111.53
Smoothing Methods: Exponential Smoothing
[([(YY22--FF22))22 + ( + (YY33--FF33))22 + ( + (YY44--FF44))22 + . . . + ( + . . . + (YY1010--FF1010))22]/9]/9
• Mean Squared Error
In order to determine which smoothing constant gives the better performance, we calculate, for each, the mean squared error for the nine weeks of forecasts, weeks 2 through 10.
Smoothing Methods: Exponential Smoothing
22334455667788991100
115115125125120120125125120120130130115115110110130130
WeekWeek = .1= .1 = .8= .8
FFtt FFtt((YYtt - - FFtt))22 ((YYtt - - FFtt))22
110.0110.000
110.5110.500
111.9111.955
112.7112.766
113.9113.988
114.5114.588
116.1116.122
116.0116.011
115.4115.411
110.0110.000
114.0114.000
122.8122.800
120.5120.566
124.1124.111
120.8120.822
128.1128.166
117.6117.633
111.5111.533
25.0025.00210.2210.2
5564.8064.80149.9149.9
4436.2536.25237.7237.7
331.261.26
36.1236.12212.8212.8
77
25.0025.00121.0121.0
007.847.84
19.7119.7116.9116.9184.2384.23173.3173.3
0058.2658.26341.2341.2
77
Sum 974.22Sum 974.22 Sum 847.52Sum 847.52Sum/9 108.25Sum/9 108.25 Sum/9 94.17Sum/9 94.17MSEMSE
YYtt
Exponential Smoothing using MINITAB
# of future forecasts needed # of data points
value for the value for the smoothing constant smoothing constant
Single Exponential Smoothing for Sales
Smoothing Constant Alpha 0.1
Time Sales Smooth Predict Error
1 110 110.000 110.000 0.0000
2 115 110.500 110.000 5.0000
3 125 111.950 110.500 14.5000
4 120 112.755 111.950 8.0500
5 125 113.980 112.755 12.2450
6 120 114.582 113.980 6.0205
7 130 116.123 114.582 15.4184
8 115 116.011 116.123 -1.1234
9 110 115.410 116.011 -6.0111
10 130 116.869 115.410 14.5901
Forecasts
Period Forecast Lower Upper
11 116.869 96.5445 137.193
Single Exponential Smoothing for Sales
Smoothing Constant Alpha 0.8
Time Sales Smooth Predict Error
1 110 110.000 110.000 0.0000
2 115 114.000 110.000 5.0000
3 125 122.800 114.000 11.0000
4 120 120.560 122.800 -2.8000
5 125 124.112 120.560 4.4400
6 120 120.822 124.112 -4.1120
7 130 128.164 120.822 9.1776
8 115 117.633 128.164 -13.1645
9 110 111.527 117.633 -7.6329
10 130 126.305 111.527 18.4734
Forecasts
Period Forecast Lower Upper
11 126.305 107.735 144.876
Week
Sale
s
1110987654321
140
130
120
110
100
Smoothing ConstantAlpha 0.1
Accuracy MeasuresMAPE 6.6994MAD 8.2958MSD 97.4232
Variable
Forecasts95.0% PI
ActualFits
Single Exponential Smoothing Plot for Sales
Weeks
Sale
s
1110987654321
150
140
130
120
110
Smoothing ConstantAlpha 0.8
Accuracy MeasuresMAPE 6.2116MAD 7.5800MSD 84.7522
Variable
Forecasts95.0% PI
ActualFits
Single Exponential Smoothing Plot for Sales
Value of the smoothing smoothing constant constant is chosen such that itminimizes the MSE